The definition of \({\operatorname{Tr}}_{{\rm Dix},\omega }\) 
extends to non-positive elements of \(\mathfrak {S}^{1^+}\)  by linearity. The
\(\omega \) -Dixmier trace
provides an example of a singular (hence non-normal) trace and it is continuous with respect to the Calderón norm (REF ), i.e.,
\(|{\operatorname{Tr}}_{{\rm Dix},\omega }(T)|\leqslant \Vert T\Vert _{1^+}\) .
A element \(T\in \mathfrak {S}^{1^+}\)  is called measurable if
the value of \(\omega (\lbrace \gamma _N(T)\rbrace _N)\)  is independent of the
choice of the generalized scale-invariant limit \(\omega \) . For a positive element \(T\geqslant 0\)  this is equivalent to the convergence of a certain Cesàro mean of \(\gamma _N(T)\)  [1]}. In particular, for a \(T\geqslant 0\)  such that \(\gamma _N(T)\)  is convergent, one has that \(T\)  is measurable and
\({\operatorname{Tr}}_{{\rm Dix}}(T) : =\lim _{N\rightarrow \infty }\left(\frac{1}{\log (N)}\sum _{n=0}^{N-1}\mu _n(T)\right) ,\) 
